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On packing spheres into containers (about Kepler's finite sphere packing problem)

Achill Schuermann
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Abstract

In an Euclidean $d$-space, the container problem asks to pack $n$ equally sized spheres into a minimal dilate of a fixed container. If the container is a smooth convex body and $d\geq 2$ we show that solutions to the container problem can not have a ``simple structure'' for large $n$. By this we in particular find that there exist arbitrary small $r>0$, such that packings in a smooth, 3-dimensional convex body, with a maximum number of spheres of radius $r$, are necessarily not hexagonal close packings. This contradicts Kepler's famous statement that the cubic or hexagonal close packing ``will be the tightest possible, so that in no other arrangement more spheres could be packed into the same container''.

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Achill Schuermann. 2006-09-09. On packing spheres into containers (about Kepler's finite sphere packing problem). https://arxiv.org/abs/math/0506200

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